To solve the quadratic equation \(4x^2 - 9x - 34 = 0\) using the quadratic formula, we start by identifying the coefficients \(a\), \(b\), and \(c\):
- \(a = 4\)
- \(b = -9\)
- \(c = -34\)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, we calculate the discriminant (\(b^2 - 4ac\)):
\[ b^2 = (-9)^2 = 81 \] \[ 4ac = 4 \cdot 4 \cdot (-34) = -544 \] \[ b^2 - 4ac = 81 - (-544) = 81 + 544 = 625 \]
Now we can substitute these values into the quadratic formula:
\[ x = \frac{-(-9) \pm \sqrt{625}}{2 \cdot 4} \]
Calculating the square root and simplifying:
\[ x = \frac{9 \pm 25}{8} \]
Now we will calculate the two possible values for \(x\):
- Using the positive sign:
\[ x_1 = \frac{9 + 25}{8} = \frac{34}{8} = 4.25 \]
- Using the negative sign:
\[ x_2 = \frac{9 - 25}{8} = \frac{-16}{8} = -2 \]
Thus, the two solutions for the equation \(4x^2 - 9x - 34 = 0\) are:
\[ x_1 = 4.25 \quad \text{and} \quad x_2 = -2 \]
According to your request to list the lesser value first, the final answer is:
\[ \boxed{-2} \quad \text{and} \quad \boxed{4.25} \]
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